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We treat the problems of phase-contrast image formation, deterministic phase retrieval and aberration balancing, in the imaging of weak objects using two-dimensional shift-invariant linear imaging systems. Three classes of model sample are considered: weak phase objects, weak phase-amplitude objects and single-material weak phase-amplitude objects. For each class of sample we show how the various aberration coefficients, which characterise a given imaging system, contribute to the structure of the associated phase-contrast image. The corresponding inverse problem, of obtaining a closed-form expression for the input wave-field given one or more aberrated phase-contrast images of the same, is then examined. Two sample applications are considered: analyser-crystal phase-contrast imaging of weak objects using hard X-rays, and Zernike-type phase-contrast imaging. We close with a discussion of how coherent and incoherent aberrations may be “balanced” against one another, briefly mentioning the applications of this idea to both “deblur by defocus” and proximity-corrected X-ray lithography.

Possibilities are investigated for quantitative computed tomography (CT) reconstruction of the spatial distribution of refractive index in a region of interest (ROI) inside an object which is larger than the field of view. The analysis is presented for the case of conventional (absorption) CT, as well as diffraction (phase-contrast) CT. It is shown that in both cases an accurate reconstruction can be achieved using the projection data corresponding to rays passing through a sufficiently wide vicinity of a ROI. Our analysis also indicates that X-ray phase-contrast CT can typically be localized to smaller regions compared to absorption CT. In particular, quasi-local hard X-ray micro-CT of regions of the order of 100 μm or even smaller in size, appears feasible with the use of propagation-based phase contrast. Numerical and experimental tests confirm the possibility of accurate quantitative CT reconstruction from truncated projections.

A new theoretical method combining analyser-based and propagation-based hard X-ray phase-contrast imaging is investigated. Unlike the previous theoretical model of the combined imaging method constructed under the assumption of slow variation of the individual transfer functions (large Fresnel numbers), a new model proposed in this paper uses the assumption of a weak scatterer (analogous to the first Born approximation). Consequently, the results are not limited to the case of short propagation distances or low-resolution imaging. An explicit expression for the combined transfer function is derived and analytical and numerical examples solving related inverse imaging problems are presented.

Analytical expressions for the spatial distribution of the spectral density and intensity of projection images are derived for a broad class of object transmission functions in the case of partially coherent Schell-model-type incident illumination. The expressions are linear with respect to the phase distribution in the transmitted wave. Associated methods for phase retrieval are discussed with the emphasis on a technique that allows simultaneous “automatic” phase retrieval and deconvolution of projection images of homogeneous objects by means of mutual cancellation at a specified defocus distance of the Fresnel diffraction effects and the image blurring due to the point-spread function of the imaging system.

A new method for rapid deblurring (partial deconvolution) is proposed which is based on the physics of coherent image formation. The method uses slightly out-of-focus images in a way that makes the Fresnel diffraction counteract the blurring due to the point-spread function. The resultant deblurring improves spatial resolution in the images and is remarkably insensitive to noise. Successful performance of several variants of this technique is demonstrated using simulated and experimental images. It is anticipated that this method will find applications in medical, astronomical and other imaging problems, as well as in lithography.

We consider the problem of quantitative phase retrieval from images obtained using a coherent shift-invariant linear imaging system whose associated transfer function (i.e., the Fourier transform of the complex point-spread function) is well approximated by a linear function of spatial frequency. This linear approximation to the transfer function is applicable when the spread of spatial frequencies, in a two-dimensional complex wavefield, is sufficiently narrow when compared to the characteristic length of variation of the transfer function for an imaging system taking such a wavefield as input. We give several algorithms for reconstructing both the phase and amplitude of a given two-dimensional coherent wavefield, given as input data one or more images of such a wavefield which may be formed by different states of the imaging system. When an object to be imaged consists of a single material, or of a single material embedded in a substrate of constant thickness, the phase-amplitude reconstruction can be performed using a single image. As a first application of these ideas, we write down an algorithm for using a single diffraction-enhanced image (DEI) to obtain a quantitative reconstruction of the projected thickness of a single-material sample which is embedded within a substrate of approximately constant thickness. This algorithm is used to quantitatively map inclusions in a breast phantom, from a single synchrotron DEI image of the same. In particular, the reconstructed images quantitatively represent the projected thickness in the bulk of the sample, in contrast to raw DEI images which greatly emphasise sharp edges (high spatial frequencies). Lastly, we point out that the methods presented here are also applicable to the quantitative analysis of differential interference contrast (DIC) images, obtained using both visible-light and X-ray microscopy.

Linear methods for solving the problem of optical phase retrieval from intensity measurements have the advantage of being deterministic and computationally more efficient compared to most alternative non-linear techniques. In the present work the relationship between two popular linear phase retrieval methods based on the Transport of Intensity equation and the first Born approximation, respectively, is clarified using a unified theoretical approach. It is also shown that a new phase retrieval method combining the two techniques has a broader validity range, higher accuracy and stability compared to the same techniques used separately. The new method is tested on simulated images and is demonstrated to be efficient for phase retrieval in the “near-Fresnel” region.

In free-space propagation of optical beams, the linear contribution of the object-plane phase to the intensity distribution in the Fresnel region is described by the Transport of Intensity equation (TIE). In the near-Fresnel region, i.e. where the Fresnel number is larger than unity, the linear term can often be dominant. Therefore in this region the TIE can provide a good initial approximation which can then be refined by other techniques, e.g. by Gerchberg–Saxton–Fienup (GSF) type algorithms. Multigrid methods with the TIE applied on coarser grids and GSF algorithms applied on finer grids are shown to be particularly effective in this context.

Methods for quantitative phase retrieval from polychromatic Fresnel-region images obtained at a fixed position along the optic axis are proposed. These methods eliminate the need for collecting images at multiple distances from the object, thus possibly simplifying the experimental procedure and allowing simultaneous recording of the image intensity data. Applications to hard X-ray in-line phase-contrast imaging with the use of conventional and energy-sensitive detectors are discussed. The suggested technique may also be applicable in quantitative phase-contrast imaging and tomography using radiation of other types than X-rays.

A method for digital phase imaging is proposed. It requires the measurement of intensity in two adjacent planes orthogonal to the optical axis. The phase is subsequently recovered by a Fast Fourier Transform of the Transport of Intensity Equation. The algorithm can be used when the illuminating beam has an arbitrary intensity distribution and is limited by a rectangular aperture. The phase retrieval formulae take on an especially simple and numerically efficient form in the case of uniform illumination. Several simulated examples are presented to confirm the viability of the algorithm.